This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Define the point on the x-axis. A point on the x-axis has a y-coordinate of 0. Let the required point be P(x, 0). The given points are A(2, -5) and B(-2, 9). Step 2: Use the distance formula. The distance formula between two points (x_1, y_1) and (x_2, y_2) is D = sqrt((x_2-x_1)^2 + (y_2-y_1)^2). Since point P is equidistant from A and B, we have PA = PB. Squaring both sides, we get PA^2 = PB^2. Step 3: Set up the equation for PA^2 = PB^2. (x-2)^2 + (0-(-5))^2 = (x-(-2))^2 + (0-9)^2 (x-2)^2 + (5)^2 = (x+2)^2 + (-9)^2 Step 4: Expand and simplify the equation. (x^2 - 4x + 4) + 25 = (x^2 + 4x + 4) + 81 x^2 - 4x + 29 = x^2 + 4x + 85 Step 5: Solve for x. Subtract x^2 from both sides: -4x + 29 = 4x + 85 Subtract 4x from both sides: -4x - 4x + 29 = 85 -8x + 29 = 85 Subtract 29 from both sides: -8x = 85 - 29 -8x = 56 Divide by -8: x = (56)/(-8) x = -7 Step 6: State the coordinates of the point. The point on the x-axis is P(x, 0), so substituting x = -7, we get the point as (-7, 0). The point on the x-axis equidistant from A(2, -5) and B(-2, 9) is (-7, 0). That's 2 down. 3 left today — send the next one.